STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS: A ROUGH PATH
VIEW
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
Abstract. We discuss regular and weak solutions to rough partial differential equations (RPDEs),
thereby providing a (rough path-)wise view on important classes of SPDEs. In contrast to many
previous works on RPDEs, our definition gives honest meaning to RPDEs as integral equation,
based on which we are able to obtain existence, uniqueness and stability results. The case of
weak “rough" forward equations, may be seen as robustification of the (measure-valued) Zakai
equation in the rough path sense. Feynman-Kac representation for RPDEs, in formal analogy to
similar classical results in SPDE theory, play an important role.
1. Introduction
Consider a diffusion process X on Rd with generator given by a second order differential operator
L. In its simplest form, the Feynman-Kac formula asserts that, for suitable data g,
(1.1)
u (t, x) = Et,x [g (XT )] ,
t ≤ T, x ∈ Rd ,
solves a parabolic partial differential equation, namely the terminal value problem
(
−∂t ut = Lut
u (T, ·) = g.
(Below we will consider slightly more general operators including zero order term, causing additional
exponential factors in the Feynman-Kac formula.) On the other hand, the law of Xt started at
X0 = x, solves the forward (or Fokker-Planck) equation
(
∂t ρt
= L∗ ρt
ρ (0, ·) = δx .
Formally at least, an infinitesimal version of (1.1) is given by
∂t hut , ρt i = h−Lut , ρt i + h ut , L∗ ρt i = 0,
and indeed the resulting duality huT , ρT i = hu0, ρ0 i is nothing than restatement of (1.1), at t = 0.
In both cases, forward and backward, there may not exist a classical C 1,2 solution. Indeed, it
suffices to consider the case of degenerate X so that ρt remains a measure; in the backward case
consider g ∈
/ C 2 . In both cases one then needs a concept of weak solutions. A natural way to
do this, consists in testing the equation in space; that is, to consider the evolution for hut , φi and
hρt , f i where φ and f are suitable test functions defined on Rd .
1991 Mathematics Subject Classification. 60H15.
Key words and phrases. stochastic partial differential equations, Zakai equation, Feynman-Kac formula, rough
partial differential equations, rough paths.
1
2
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
Applications from filtering theory lead to (backward) SPDEs of the form
(
−dut
= L[ut ]dt + Γ[ut ] ◦ dWt
u (T, ·) = g,
where W = (W 1 , . . . , W e ) and Γ = (Γ1 , . . . , Γe ) are first order differential operators,1 in duality
with the forward (or Zakai) equation
(
dρt
= L∗ [ρt ]dt + Γ∗ [ρt ] ◦ dWt
ρ (0, ·) = δx .
Such SPDEs were studied extensively in classical works [29, 33, 31]. It is a natural question, studied
for instance in a series of papers by Gyöngy [23, 24], to what extent such SPDEs are approximated
by (random) PDEs, upon replacing the (Stratonovich) differential dW = dW (ω) by Ẇ ε (ω) dt, given
a suitable family of smooth approximation (W ε ) to Brownian motion. In recent works [17, 11], also
[19, Ch.12], it was shown that the backward solutions uε , interpreted as viscosity solution (assuming
g ∈ Cb ) actually converge locally uniformly, with limit u only depending on the rough path limit
of (W ε ). Writing W = (W , W) for such a (deterministic!) rough path (see e.g. [19] for notation)
say, α-Hölder, for 1/3 < α < 1/2) the question arises if one can give an honest meaning to the
equations
(1.2)
−dut
=
L[ut ]dt + Γ[ut ]dWt ,
dρt
=
L∗ [ρt ]dt + Γ∗ [ρt ]dWt .
In the aforementioned works, these “rough partial differential equations" (RPDEs) had only formal
meaning. The actual definition was then given either in terms of a (flow)transformed equation in
the spirit of Kunita (e.g. [17], also [19, p.177]) or in terms of a unique continuous extension of the
PDE solution as function of driving noise, [5, 17].
There are two difficulties with such rough partial differential equations. The first one is the
temporal roughness of W, a problem that has been well-understood from the rough path analysis
of SDEs. Indeed, following Davie’s approach to RDEs [8], the (rough) pathwise meaning of
dX = β (X) dW
is, by definition, and writing Xs,t = Xt − Xs for path increments,
Xs,t = β (Xs ) Ws,t + β 0 β (Xs ) Ws,t + o (|t − s|) .
Under suitable assumptions on β, uniqueness, local/global existence results are well-known. This
quantifies that statement that X is controlled by W , with “Gubinelli derivative" β (X), and in turn
implies the integral representation in terms of a bona-fide rough integral (cf. [19, Ch.4])
Z t
X
Xt − Xs =
β (X) dW = lim
β (Xu ) Wu,v + β 0 β (Xu ) Wu,v .
s
[u,v]∈P
This suggests that the meaning of the backward equation (1.2) is
Z t
Z t
u (s, x) − u (t, x) =
L[ur ]dr +
Γ[ur ]dWr ,
s
1Write Γ[u] ◦ dW =
Pe
k=1
Γk [u] ◦ dW k .
s
SPDES AND ROUGH PATHS
3
provided u is sufficiently regular (in space) such as to make L[u], Γ[u] meaningful, and provided the
last term makes sense as rough integral. The other difficulty is exactly that u may not be regular in
space so that L[u], Γ[u] require a weak meaning. More precisely, we propose the following spatially
weak2 formulation, of the form
Z t
Z t
hus , φi − hut , φi =
hur , L∗ φi dr +
hur , Γ∗ φi dWr ,
s
s
where, again, we can hope to understand the last term as rough integral. (Everything said for
backward equations translates, mutatis mutandis, to the forward setting.)
The main result of this paper is that - in all cases - one has existence and uniqueness results.
Loosely speaking (and subject to suitable regularity assumptions on the coefficients of L, Γ; but no
ellipticity assumptions) we have
Theorem 1. For nice terminal data g there exists a unique (spatially) regular solution to the
backward RPDE. Similarly, for nice initial data ρ0 (with nice density p0 , say) the forward RPDE
has a unique (spatially) regular solution.
If the terminal data g is only bounded and continuous, we have existence and uniqueness of a weak
solution to the backward RPDE. Similarly, if the initial data ρ0 of the forward RPDE is only a
finite measure, we have existence and uniqueness of a weak (here: measure-valued) solution to the
forward RPDE.
In all cases, the (unique) solution depends continuously on the driving rough path and we have
Feynman–Kac type representation formulae.
Let us briefly discuss the strategy of proof. In all cases (regular/weak, forward/backward)
existence of a solution is verified via an explicit Feynman–Kac type formula, based on a notion
of “hybrid" Itô/rough differential equation, which already appeared in previous works [9, 11], see
also [19]. We then use regular forward existence to show weak backward uniqueness (Theorem
9), which actually requires us to work with exponentially decaying test functions. Next, regular
backward existence leads to weak (actually, measure-valued) forward uniqueness (Theorem 16), here
we just need boundedness and some control in the sense of Gubinelli. Then weak (measure-valued)
forward existence gives regular backward uniqueness. At last, we note that, subject to suitable
smoothness assumptions on the coefficients, regular forward equations can be viewed as regular
backward equations, from which we deduce regular forward uniqueness.
It is a natural question what the above RPDE solutions have to do with classical SPDE solutions.
To this end, recall [19, Ch.9] consistency of RDEs with SDEs in the following sense: RDE solutions
driven by W = WStrato (ω), the usual (random) geometric rough path associated to Brownian
motion W via iterated Stratonovich integration are solutions to the corresponding (Stratonovich)
SDEs. Consider now - for the sake of argument - a regular backward RPDE solution; that is, the
unique solution u = u (t, x; W) to
−dut = L[ut ]dt + Γ[ut ]dWt
(with fixed
(1.3)
Cb2
time-T terminal data). We expect that
ũ (t, x) = ũ (t, x; ω) = u t, x; WStrato (ω)
2There is no probability here, for W is a deterministic rough path. Nevertheless, with a view to later applications
to SPDEs and to avoid misunderstandings, let us emphasize that in this paper “weak" is always understood as
“analytically weak".
4
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
is also a (and hopefully: the unique) solution to the (backward) SPDE, again with fixed terminal
data,
−dũt = L[ũt ]dt + Γ[ũt ] ◦ dWt .
(Similar for weak backward and weak/regular forward equations.) Unfortunately, we cannot hope
for a general RPDE/SPDE consistency statement for the simple reason that the choice of spaces in
which SPDE existence and uniqueness statements are proven are model-dependent and therefore
vary from paper to paper. In other words, checking that ũ (t, x; ω) is a - and then the (unique) SPDE solution within a given SPDE setting will necessarily require to check details specific to this
setting. Luckily, there are arguments which do not force us into such a particular setting.
• Consider a notion of (Stratonovich) SPDE solution for which there are existence, uniqueness results and Wong–Zakai stability, by which we mean that the (unique bounded, or
finite-measure valued) solutions to the random PDEs obtained by replacing dW (ω) by the
mollified Ẇ ε (ω) dt converge to the unique SPDE solution. (Such Wong–Zakai results are
found e.g. in the works of Gyöngy.) Assume also that our regularity assumptions fall within
the scope of these existence and uniqueness results. Then, for fixed terminal (resp. initial)
data, our unique RPDE solution, with driving rough path W = WStrato (ω), coincides with
(and in fact, maybe a very pleasant version of) the unique SPDE solution. (This follows immediately from continuous dependence of our RPDE solutions on the driving rough paths,
together with well-known rough path convergence of mollifier approximations [14].) In a
context of viscosity solutions, this argument was spelled out in [17].
• Consider a notion of (Stratonovich) SPDE solution for which there are existence, uniqueness
results and a Feynman–Kac representation formula. (This is the case in essentially every
classical work on linear SPDEs, especially in the filtering context.) Recall that such SPDE
Feynman–Kac formulas are conditional expectations, given W (ω) (the observation, in the
filtering context). In contrast, the Feynman–Kac formula eluded to in Theorem 1, is of
unconditional form Et,x (...), the expectation taken over some hybrid Itô-rough process
(with rough driver dW). By a stochastic Fubini argument (similar to the one in [11]) one
can show that the Feynman–Kac formula, evaluated at W = WStrato (ω), indeed yields
the SPDE Feynman–Kac formula. In particular, our unique RPDE solution, with driving
rough path W = WStrato (ω), then coincides with the unique SPDE solution.
• At last, we consider an immediate consequence of our (rough path-) wise definition in case
of W = WStrato (ω). For the sake of argument, let us now focus on the weak backward
equation,
Z t
Z t
hus , φi − hut , φi =
hur , L∗ φi dr +
hur , Γ∗ φi dWr .
s
s
With ũ (t, x; ω) = u t, x; W
(ω) , as before it follows from consistency of rough with
classical (backward) Stratonovich integration [19, Ch.5] that
Z t
Z t
hũs , φi − hũt , φi =
hũr , L∗ φi dr +
hũr , Γ∗ φi ◦ dW,
Strato
s
s
for the same class of spatial test functions. Such notion of weak (or distributional) SPDE
solutions appear for instance in the works of Krylov, e.g. [25, Def. 4.6]. Hence, whenever
such a notion of SPDE solution comes with uniqueness results, it is straight-forward to see
that ũ, i.e. our solution constructed via rough paths, must coincide with the unique SPDE
solution.
SPDES AND ROUGH PATHS
5
1.1. Notation. The second resp. first oder operators we shall consider are of the following form,
1
Tr σ (x) σ T (x) D2 u + hb (x) , Dui + c (x) u
Lu :=
2
Γk u := hβk (x) , Dui + γk (x) u;
with σ = (σ1 , . . . , σdB ) , β = (β1 , . . . , βe ) and b vector fields on Rd and scalar functions c, γ1 , . . . , γe .
We note that the formal adjoints are given as,
1
Tr[ã(x)D2 ϕ] + hb̃(x), Dϕi + c̃(x)ϕ,
2
Γ∗k ϕ = hβ̃k (x), Dϕi + γ̃k (x)ϕ
L∗ ϕ =
where
ã(x) := a(x) := σσ T (x)
β̃k (x) := −βk (x)
b̃i (x) := ∂j aji (x) − bi (x)
γ̃k (x) := − div(βk )(x) + γk (x)
1
c̃(x) := ∂ij aij (x) − div(b)(x) + c(x)
2
Precise assumptions on the coefficients will appear in the theorems below. Let us remark,
however, that we did not push for optimal assumptions. As is typical in rough path theory, Cbn regularity (bounded, with bounded derivatives up to order n) can often be improved to Cbγ -regularity
with γ ∈ (n − 1, n), depending on the Hölder exponent of the driving rough path.
(1.4)
2. The backward equation
Replacing the rough path by a smooth path, say W ∈ C 1 ([0, T ] , Re ) we certainly want to recover
a solution to the PDE
(
−∂t ut = Lut + Γk ut Ẇtk
(2.1)
u (T, ·) = g.
For the precise statement of the following lemma, let us now introduce a suitable class of test
functions with exponential decay, that will become important in the concept of weak solutions.
n
(Rd ) the class of functions φ ∈ C n (Rd ) such that there
Definition 2. For n ≥ 0 denote with Cexp
exists c > 0 such that
1
|Dk φ(x)| ≤ ce− c |x| , k = 0, 1, . . . , n.
Define the quasinorm3 || · ||Cexp
n (Rd ) as the infimum over the values of c satisfying the bound. Define
m,n
moreover the space Cexp
([0, T ] × Rd ) to be the class of functions φ ∈ C m,n ([0, T ] × Rd ) such that
there exists c > 0 such that
1
|Dj,k φ(t, x)| ≤ ce− c |x| , j = 0, . . . , m, k = 0, 1, . . . , n.
We then recall the following Feynman-Kac representation for solutions to the classical equation
(2.1).
3 .. which we shall need in order to speak of bounded sets in C n (Rd ) ..
exp
6
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
Lemma 3. Assume c, b, σi , γj , βk ∈ Cb2 , i = 1, . . . , dB , j, k = 1, . . . , e. Let u be given as
"
!#
Z T
Z T
t,x
c (Xr ) dr +
γ (Xr ) Ẇr dr
(2.2)
u (t, x) = E
g (XT ) exp
t
t
with
dXt = σ (Xt ) dB (ω) + b (Xt ) dt + β (Xt ) Ẇt dt,
where B is a dB -dimensional Brownian motion and W ∈ C 1 ([0, T ], Re ).
2
(i) If g ∈ Cb2 (Rd ) then u is the unique Cb1,2 ([0, T ]×Rd ) solution to (2.1). If moreover g ∈ Cexp
(Rd )
1,2
d
then u ∈ Cexp ([0, T ] × R ).
(ii) If g ∈ Cb (Rd ) then u ∈ Cb ([0, T ] × Rd ) and it is the unique bounded analytically weak solution
to (2.1), that is, for ϕ ∈ D(Rd )
Z T
Z T
∗
hut , ϕi = hg, ϕi +
hur , L ϕi dr +
(2.3)
hur , Γ∗ ϕi dWr .
t
t
Proof. Let us first note that the expectation actually exists, since g, c, γ and |Ẇ | are bounded.
(i): The proof amounts to taking derivatives under the expectation, see for example Theorem
V.7.4 in [26], which shows that u is a Cb1,2 solution.4 Uniqueness follows from the maximum
principle, see for example Theorem 8.1.4 in [27].
1,2
2
([0, T ] × Rd ). This is similar to the
(Rd ) then one can show that actually u ∈ Cexp
If g ∈ Cexp
rough case in Theorem 9, so we omit the proof here.
(ii): Take some g n ∈ Cb2 (Rd ) converging to g locally uniformly, uniformly bounded by 2||g||∞
Let un be the corresponding classical solution from part (i). Then un satisfies (2.3) with g replaced
by g n . Now by the Feynman-Kac representation, we get for every N > 0,
|un (t, x) − u(t, x)| . E[|g n (XTt,x ) − g(XTt,x )|2 ]1/2
≤ sup |g n (y) − g(y)| + 2||g||∞ E[1[−N,N ]C (|XTt,x )].
|y|≤N
Hence for every R > 0
sup |un (t, x) − u(t, x)| . sup |g n (y) − g(y)| +
|x|≤R
|y|≤N
1
sup E[|XTt,x |],
N |x|≤R
from which the locally uniform convergence of unt to ut follows, uniformly in t ≤ T . Taking the
limit in the integral equation, we then see that u satisfies (2.3).
To show uniqueness, let u ∈ Cb ([0, T ] × Rd ) be any solution to (2.3). It is immediate that the
equation then also holds for test functions ϕ ∈ Cc2 (Rd ). It is straightforward to show that for
ϕ ∈ Cc1,2 ([0, T ] × Rd ) we have
Z T
Z T
(2.4)
hut , ϕt i = hg, ϕT i +
hur , −∂t ϕr + L∗ ϕr i dr +
hur , Γ∗ ϕr i dWr .
t
Finally, via dominated convergence, (2.4) also holds for ϕ ∈
t
1,2
Cexp
([0, T ]
× Rd ).
4 In [26] it is assumed that the term in the exponential is non-positive, but a term bounded from below poses no
additional difficulty: just replace u(t, x) by u(t, x)e−c(T −t) for c sufficiently large.
SPDES AND ROUGH PATHS
7
4
Now Lemma 12 (iv) gives us for every t ∈ [0, T ), φ ∈ Cexp
(Rd ) such a ϕ (on [t, T ]) that satisfies
∂s ϕs = L∗ ϕs + Γ∗ ϕs Ẇs
ϕt = φ.
Then, by (2.4),
hut , φi = hut , ϕt i = hg, ϕT i.
So, tested against φ ∈ Cc4 (Rd ), all solutions coincide at every t ∈ [0, T ], which gives uniqueness in
Cb ([0, T ] × Rd ).
When replacing W by a rough path W, we are formally interested in the following equation
(
−du
= Ludt + Γk udWk
.
(2.5)
u (T, ·) = g.
We will next introduce two solution concepts, weak and regular in nature (see Definitions 4 and
7 below).
Definition 4 (analytically weak backward RPDE solution). Given an α-Hölder rough path
W = (W, W), α ∈ (1/3, 1/2], we say that a bounded, measurable function u = u (t, x; W) =
3
ut (x; W) is an analytically weak solution to (2.5), if for all functions ϕ ∈ Cexp
(Rd ), we have
ϕ
ϕ 0
2α
(Y , (Y ) ) ∈ DW with
Ytϕ := hut , Γ∗i ϕi ∈ Re , (Ytϕ )0 := − ut , Γ∗j Γ∗i ϕ ∈ L(Re , Re ),
that is
(2.6)
||Y ϕ , (Y ϕ )0 ||W,α < ∞,
and the following equation is satisfied
Z T
Z T
(2.7)
hut , ϕi = hg, ϕi +
hur , L∗ ϕi dr +
hur , Γ∗ ϕi dWr ,
t
t
R
Here, Y dW is the rough integral against (Y, Y 0 ).
0 ≤ t ≤ T.
Remark 5. Different from the smooth case, Lemma 3, we work with test functions in the larger
3
class Cexp
here. This is necessary, since the presence of the rough integral makes it impossible to
automatically enlarge the space of functions for which the integral equation holds, as was done in
the proof of Lemma 3.
Remark 6. Heuristically, the origin of the compensator term Yt0 = hut , Γ∗ Γ∗ ϕi can be seen as
follows. One certainly expects that
Z t
hur , Γ∗ ϕi dWr ≈ hus , Γ∗ ϕi Ws,t
s
2α
. Hence, in view of (2.7),
where a ≈ b means a − b = O |t − s|
Z t
hut , ϕi − hus , ϕi ≈ −
hu, Γ∗ ϕi dW ≈ − hus , Γ∗ ϕi Ws,t
s
3
Replacing ϕ by Γ∗ ϕ (note that the latter is not in Cexp
though) gives
2α
hut , Γ∗ ϕi − hus , Γ∗ ϕi = − hus , Γ∗ Γ∗ ϕi Ws,t + O |v − u|
8
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
so that t 7→ hut , Γ∗ ϕi is controlled by W , with Gubinelli derivative − hut , Γ∗ Γ∗ ϕi.
Definition 7 (regular backward RPDE solution). Given an α-Hölder rough path W = (W, W),
α ∈ (1/3, 1/2], we say that a function u = u (t, x; W) ∈ C 0,2 (with respect to t, x) is a solution to
(2.5) if (Γk u, Γj Γk u) is controlled by W and
Z T
Z T
Γk u(r, x)dWrk .
Lu(r, x)dr +
u(t, x) = g(x) +
t
t
Remark 8. If a regular solution in the sense of Definition 7 possesses a uniform bound on the
control (see for example (2.9) below) then it is also a weak solution in the sense of Definition 4.
Theorem 9. Throughout, W is a geometric α-Hölder rough path, α ∈ (1/3, 1/2]. Assume σi , βj ∈
Cb3 (Rd ), b ∈ Cb1 (Rd ), c ∈ Cb1 (Rd ), γk ∈ Cb2 (Rd ). Consider g ∈ Cb0 (Rd ).
(i) Stability. Let u = uW be the solution to (2.1) as given by the Feynman-Kac representation
(2.2), whenever W ∈ C 1 . Pick W ∈ C 1 convergent in rough path sense to W. Then there
exists a bounded, continuous function uW , independent of the choice of the approximating se
quence, so that uW → uW uniformly. The resulting map W 7→ uW is continuous. Moreover,
the following Feynman-Kac representation holds,
"
!#
Z T
Z T
W
t,x
u (t, x) = E
g (XT ) exp
c (Xr ) dr +
γ (Xr ) dWr
,
t
t
where X solves the rough SDE (see Appendix, Lemma 34)
dXt = σ (Xt ) dB (ω) + b (Xt ) dt + β (Xt ) dWt ,
(2.8)
where B is a dB -dimensional Brownian motion.
(ii) Analytically weak backward RPDE solution. Let u = uW be the function constructed
in (i). Then u = uW ∈ Cb ([0, T ]×Rd ) is a bounded solution to (2.5) in the sense of Definition
3
4. Moreover, (2.6) is bounded, uniformly over bounded sets of ϕ in Cexp
(Rd ), and it is the
only solution in the class of Cb functions u satisfying (2.6).
(iii) Analytically regular backward RPDE solution. Assume σi , βj ∈ Cb6 (Rd ), b ∈ Cb4 (Rd ), c ∈
Cb4 (Rd ), γk ∈ Cb6 (Rd ) and g ∈ Cb4 (Rd ). Then u = uW ∈ Cb0,4 ([0, T ] × Rd ) is a bounded solution to (2.5) in the sense of Definition 7. It is the only solution in the class of functions in
Cb0,4 ([0, T ] × Rd ) that satisfy
sup ||Γu(·, x), ΓΓu(·, x)||W,α < ∞.
(2.9)
x
If moreover g ∈
4
Cexp
(Rd ),
0,4
then u ∈ Cexp
([0, T ] × Rd ).
Remark 10. We consider solutions in Cb0,4 , instead of the obvious choice Cb0,2 , because of two
reasons. First, in order to show that u is controlled by W we need g ∈ Cb4 (Rd ) which automatically
gives us u ∈ Cb0,4 ([0, T ] × Rd ). Second, this additional regularity is needed for the uniqueness proof
via duality.
Remark 11. Results of the type in Theorem 9 (i), even in nonlinear situations, were obtained in
[4, 5, 10, 9, 17]. However, in all these references, the only intrinsic meaning of these equations was
given in terms of a transformed equation, somewhat in the spirit of the Lions-Souganidis [30] theory
of stochastic viscosity solutions. On the contrary, part (ii) and (iii) of the above theorem present a
direct intrinsic characterization. See also [7, Chapter 3].
SPDES AND ROUGH PATHS
9
Proof. (i) This follows from stability of “rough SDEs”, see Lemma 34.
(ii) Existence For simplicity only, we take c = γ = b = 0 so that
u (s, x)
=
E [g (XTs,x )] ,
dXt
=
σ (Xt ) dBt (ω) + β (Xt ) dWt .
s,x
(With X we mean the unique solution started at Xs = x.) In the following we consider the above
SDE as an RDE w.r.t. the joint lift Z = (Z, Z) of W and the Brownian motion B (see Lemma 33
and Lemma 34 below). Denote with Φ its associated flow.
3
Recall Yt = hut , ϕ̄i, Yt0 = −hut , Γ∗ ϕ̄i, where ϕ̄ := Γ∗ ϕ. Since ϕ ∈ Cexp
and βj ∈ Cb2 , j = 1, . . . , e,
2
we have that ϕ̄ ∈ Cexp . Then
Yt − Ys − Yt0 Ws,t
Z
=E
{g(Φt,T (x)) − g(Φs,T (x))} ϕ̄(x) + g(Φt,T (x))Γ∗ ϕ̄(x)Ws,t dx
Rd
hZ
n
−1
−1
−1
=E
g(y) ϕ̄(Φ−1
t,T (y)) det(DΦt,T (y)) − ϕ̄(Φs,T (y)) det(DΦs,T (y))
Rd
o i
−1
+ Γ∗ ϕ̄(Φ−1
(y))
det(DΦ
(y))W
dy
s,t
t,T
t,T
n
hZ
−1
−1
−1
g(y) ϕ̄(Φ−1
=E
t,T (y)) det(DΦt,T (y)) − ϕ̄(Φs,T (y)) det(DΦs,T (y))
Rd
o i
−1
−1
−1
∗
+ Γ∗ ϕ̄(Φ−1
(y))
det(DΦ
(y))W
+
Γ
ϕ̄(Φ
(y))
det(DΦ
(y))B
dy .
s,t
s,t
t,T
t,T
t,T
t,T
Note that Γ∗ ϕ = − div (bϕ). Hence the term in curly brackets is bounded in absolute value, using
Lemma 24, by a constant times
||ϕ̄||Cb3 (M (y)) exp(CN1;[0,T ] (Z)) (||Z||β + 1)
17+3d
|t − s|2α .
Hence
|Yt − Ys − Yt0 Ws,t | .
Z
Rd
h
i
17+3d
E ||ϕ||Cb3 (M (y)) exp(CN1;[0,T ] (Z)) (||Z||β + 1)
dy |t − s|2α .
Next observe that
h
i
17+3d
E ||ϕ||Cb3 (M (y)) exp(CN1;[0,T ] (Z)) (||Z||β + 1)
h
i1/2 h
i1/2
17+3d
≤ E ||ϕ||3C 2 (M (y))
E exp(CN1;[0,T ] (Z)) (||Z||β + 1)
,
b
and Lemma 33 now implies that the last term is bounded and Lemma 36 implies that the first term
decays exponentially in y. Therefore
|Yt − Ys − Yt0 Ws,t | . |t − s|2α ,
as desired.
The estimate |Yt0 − Ys0 | ≤ C|t − s|α is shown analogously, and then
0
|Yt − Ys − Ys0 Ws,t | ≤ |Yt − Ys − Yt0 Ws,t | + |Ys,t
Ws,t | = O(|t − s|2α ),
as desired.
It remains to show that the integral equation (2.7) is satisfied. For this let W n be a sequence of
smooth paths converging to W in α-rough path metric. Let un be the solution to (2.3) as given by
Lemma 3 (ii).
10
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
Part (i) of the theorem now implies that un converges locally uniformly to u, hence the convergence of all the terms in (2.7) except the rough integral is immediate. For the rough integral, in
view of Theorem 9.1 in [19], it is enough to show that
n
sup ||Y 0 − Y 0 ||∞ → 0
sup ||Rn ||2α < ∞
sup ||Rn − R||∞ → 0,
sup ||Y 0 ||α < ∞
n
n
n
with
Ytn
:=
n
hunt , Γ∗ ϕi, Y 0
:=
n
n
hunt , Γ∗ Γ∗ ϕi
and
n
n
Rs,t
= hunt − uns , Γ∗ ϕi − huns , Γ∗ Γ∗ ϕiWs,t
.
The first two statements follow from the fact that the preceding considerations were uniform for W
n
bounded in rough path norm. Finally, convergence in supremum norm of Y 0 t −Yt0 = hunt −ut , Γ∗ Γ∗ ϕi
n
n
and Rs,t − Rs,t follows from local uniform convergence of u .
0,3
Uniqueness Let φ ∈ Cexp
([t, T ], Rd ) be such that
Z T
Z T
φ(t, x) = ϕ(x) +
α(r, x)dr +
ηi (r, x)dWri ,
t
with η ∈
δ>0
0,3
([0, T ]
Cexp
d
× R ), and
t
0
(ηi=1,...,e , ηi,j=1,...,e
)
controlled by W . Assume moreover for some
||η(x), η 0 (x)||W,α . e−δ|x| ,
||Dη(x), Dη 0 (x)||W,α . e−δ|x| .
Then by Lemma 32
Z
huT , φT i = hut , φt i −
T
∗
Z
hur , L φr idr −
t
T
hur , Γ∗k φr idWrk
Z
T
hur , α(r)idr +
+
t
Z
t
∗
T
hur , ηk (r)idWrk .
t
Γ∗i φ(r)
0
So it remains to find, for given ϕ, such a φ with α(r) = L φ(r), ηi (r) =
and ηi,j
(r) =
4
d
But this is exactly what Theorem 16 (iii) gives us for ϕ ∈ Cexp (R ). Then
Γ∗j Γ∗i φ(r).
hg, φT i = huT , φT i = hut , φt i = hut , ϕi,
which gives uniqueness of ut . This holds for all t ∈ [0, T ], which gives uniqueness of u.
(iii) Again, for simplicity only, we take c = γ = b = 0 so that
u(t, x) = E g XTt,x .
Then
Du(t, x) = E[Dg(XTt,x )DXTt,x ].
indeed, using the integrability of DX given by Lemma 34 and the fact that Dg is bounded, the
statement follows from interchanging differentiation and integration, see for example Theorem 8.1.2
in [12].
SPDES AND ROUGH PATHS
11
Then by Lemma 29
Du(t, x) − Du(s, x)
= E[Dg(XTt,x )DXTt,x − Dg(XTs,x )DXTs,x ]
Z t
Z t
r,x
r,x
r,x
r,x
r,x
2
Dg(XT )dDXT +
D g(XT )hdXT , DXT ·i
=E
s
s
Z t
hZ t
r,x
2 r,x
Dg(XTr,x )DV (x)DXTr,x dZr
=E
Dg(XT )D XT hV (x)dZr , ·i +
s
s
Z t
i
+
D2 g(XTr,x )hDXTr,x V (x)dZr , DXTr,x ·i + O(|t − s|)
s
= E[Dg(XTs,x )D2 XTs,x hβ(x)Ws,t , ·i] + E[Dg(XTs,x )DXTs,x Dβ(x)]Ws,t
+ E[D2 g(XTs,x )hDXTs,x V (x)Ws,t , DXTs,x ·i] + O(|t − s|2α )
= ∂x [β(x)Du(s, x)]Ws,t + O(|t − s|2α )
So Γu is controlled as claimed and (2.9) is satisfied. Showing that u ∈ Cb0,4 also follows from
differentiation under the expectation and the proof that the integral equation is satisfied now
follows by using smooth approximations to W, as in part (ii).
Uniqueness follows from existence of the measure-valued forward equation. The argument is dual
to the one that will be used in the proof of Theorem 16 (ii), so we omit the proof here.
4
Finally, the exponential decay of u, if g ∈ Cexp
, follows from Lemma 35.
3. The forward equation
We now consider the forward equation
(
∂t ρt
(3.1)
ρ0
= L∗ ρt + Γ∗k ρt Ẇtk
= p0 .
on the space M(Rd ) of finite measures on Rd .
Equation (3.1) is dual to the backward equation - considered in the previous section - in a sense
that will be made precise in the following (see in particular Corollary 18 below).
The space M(Rd ) is endowed with the weak topology; that is µn → µ if µn (f ) → µ(f ) for all
f ∈ Cb (Rd ). It is metrizable with compatible metric given by the Kantorovich-Rubinstein metric
d, defined as
Z
Z
d(µ, ν) :=
sup
f (x)ν(dx) −
f (x)µ(dx)
||f ||C 1 (Rd ) ≤1
b
Rd
Rd
(see Chapter 8.3 in [2]). A compatible metric on the space of continuous finite-measure-valued
paths is then given by d(µ· , ρ· ) := supt≤T d(µt , ρt ).
Lemma 12. Assume c, b, σi , γj , βk ∈ Cb2 , i = 1, . . . , dB , j, k = 1, . . . , e. Define for W ∈ C 1 the
measure valued process ρ via its action on f ∈ Cb (Rd ) as
Z t
Z s
(3.2)
ρt (f ) := E0,ν f (Xt ) exp
c (Xs ) ds +
γ (Xs ) Ẇs ds ,
0
0
12
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
where ν ∈ M(Rd ) is the initial condition of the diffusion X with dynamics
dXt = σ (Xt ) dB (ω) + b (Xt ) dt + β (Xt ) Ẇt dt,
where B is a dB -dimensional Brownian motion.
(i) Then ρ is the unique, continuous M(Rd )-valued path which satisfies, for all f ∈ Cb2 (Rd ),
Z t
Z t
(3.3)
ρs (Γk f )dWsk .
ρs (Lf )ds +
ρt (f ) = ν(f ) +
0
0
(ii) Assume moreover σi ∈ Cb4 (Rd ), i = 1, . . . , dB , b, βk ∈ Cb3 (Rd ), k = 1, . . . , e. If ν has a density
2
1,2
p0 ∈ Cexp
(Rd ) then ρt has a density p ∈ Cexp
([0, T ]×Rd ) which is the unique bounded classical
solution to (3.1).
2
Remark 13. We choose p0 ∈ Cexp
(Rd ) in part (ii) since this is what we shall work with in the
rough case. In the smooth case, the assumptions on the density p0 can be weakened. Assume for
example that ν has a density p0 ∈ Cb2 ∩ L1 . Then ρt has a Cb2 density pt for all t ≥ 0 and p ∈ Cb1,2
is the unique bounded classical solution to (3.1). Moreover,
Z t
Z t
(3.4)
||pt ||L1 (Rd ) = ρt (1) = E0,ν exp
c (Xs ) ds +
γ (Xs ) Ẇs ds .
0
0
Indeed, by the smoothness assumptions on the coefficients, (3.1) has a unique solution in Cb1,2
(this can again be seen by a Feynman-Kac argument, as in Lemma 3).
We have to show that the unique classical solution pt ∈ Cb2 of (3.1) with non-negative initial
condition p0 ∈ Cb2 ∩ L1 is integrable. First recall that from the maximum principle,
pt R≥ 0 for all
R
d
t ≥ 0 (see for example Theorem 8.1.4 in [27]). Note that (3.1) implies that dt
ϕpt dx = L̃t ϕpt dx,
where L̃t φ := Lφ + Γk φẆtk , hence
Z
Z
Z tZ
L̃s ϕps dx ds
(3.5)
ϕpt dx = ϕp0 dx +
0
for any smooth and compactly supported function ϕ. Our aim now is to extend this equality to the
constant function ϕ ≡ 1. To this end consider for ε > 0 the function
ϕε (x) := ϕ εkxk2 ,
where ϕ(r) = (1 + r)−
e+1
2
, r ≥ 0. It is easy to check that both ϕε and


X
X
e+3
−
2
 (σσ T )ij (x) +
Lϕε (x) = −ε(e + 1) 1 + εkxk2
(b̃i )t (x)xi  + cϕε (x)
ij
i


X
e+5
−
2
 (σσ T )ij (x)xi xj 
+ ε2 (e + 1)(e + 3) 1 + εkxk2
ij
are integrable. Since the coefficients σσ T and b̃t := b + Γk Ẇtk have at most linear growth and c is
bounded, there exists a finite constant M , independent of ε, such that
L̃s ϕε ≤ M ϕε .
SPDES AND ROUGH PATHS
13
Next fix a smooth
2 compactly supported test function χ on R satisfying 1[−1,1] ≤ χ ≤ 1[−2,2] and
let χN (x) := χ kxk
N 2 . Then χN ϕε is compactly supported and
d+3 X
e + 1 0 kxk2
2 − 2
L̃t (χN ϕε ) = χN L̃t ϕε − 4ε 2 χ
1
+
εkxk
(σσ T )(x)xi xj
N
N2
ij
+ ((L0 )t χN )ϕε
where (L̃0 )t u = L̃t u − cu. Again due to the assumptions on the coefficients of L (resp. L0 ) we
obtain that L0 χN is uniformly bounded in N , so that |L̃t (χN ϕε ) | is uniformly bounded in N in
terms of ϕε and |L̃t ϕε |. Since L̃t (χN ϕε ) → L̃t ϕε pointwise, Lebesgue’s dominated convergence now
implies that (3.5) extends to the limit N → ∞, hence
Z
Z tZ
Z
ϕε pt dx =
ϕε p0 dx +
L̃s ϕε ps dx ds
Z tZ
ϕε p0 dx + M
ϕε ps dx ds .
0
(3.6)
Z
≤
0
Gronwall’s lemma now implies that
Z
ϕε pt dx ≤ eM t
Z
ϕε p0 dx .
Since p0 is integrable, we can now take the limit ε ↓ 0 to conclude with Fatou’s lemma that
Z
Z
pt dx ≤ eM t p0 dx < ∞ .
R
Hence νt (f ) := pt (x)f (x)dx defines a finite-measure valued path and it satisfies (3.3). By
uniqueness it hence coincides with ρ. The expression for the L1 -norm of pt then follows from (i).
Proof. (i): Equation (3.3) is satisfied by an application of Itō’s formula, see for example Theorem
3.24 in [1] for a similar argument. Uniqueness follows as in Theorem 4.16 in [1]. Let us sketch the
argument.5 First one shows that every solution to (3.3) also satisfies for ϕ ∈ Cb1,2 ([0, T ], Rd )
Z t
Z t
ρt (ϕ) = ν(ϕ) +
ρs (∂t ϕ + Lϕ)ds +
(3.7)
ρs (Γk ϕ)dWsk .
0
Given now Φ ∈
equation
Cb∞ (Rd )
0
and t ≤ T consider any solution v ∈ Cb1,2 ([0, t], Rd ) to the backward
−∂t v = Lv + Γk v Ẇtk
v(t, ·) = Φ,
the existence of which follows from Lemma 3.
Given two solutions ρ, ρ̄ to (3.3), we then have, by (3.7),
ρt (Φ) = ρt (vt ) = ρ0 (v0 ) = ρ̄0 (v0 ) = ρ̄t (vt ) = ρ̄t (Φ),
Cb∞ .
so ρt , ρ̄ coincide on
By pointwise uniformly bounded convergence they then also coincide on
Cb , and hence ρt = ρ̄t as desired.
5 Our setting here is simpler then in [1], since our coefficients and their derivatives are bounded.
14
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
(ii): The coefficients of the dual equation, see (1.4), all are in Cb2 . Hence using Lemma 3
1,2
there exists a unique bounded classical solution p ∈ Cexp
([0, T ] × Rd ) to the PDE and a backward
Feynman-Kac representation holds. Then it is in particular integrable and hence defines a measurevalued function µ on [0, T ] which satisfies (3.3). By uniqueness for this equation it coincides with
ρ.
When replacing W by a rough path W, we are interested in the following equation
(
dρt = L∗ ρt dt + Γ∗k ρt dWtk
(3.8)
ρ0 = ν.
Two ways to make sense of this equation are given in the following definitions.
Definition 14 (Measure valued forward RPDE solution). Given an α-Hölder rough path
W = (W, W), α ∈ (1/3, 1/2], and ν ∈ M(Rd ), we say that a continuous finite-measure-valued path
ρt is a weak solution to (3.8) if for all f ∈ Cb3 (Rd ), ρt (Γk f )k=1,...,e is controlled by W with Gubinelli
derivative ρt (Γj Γk f )k,j=1,...,e , that is
||ρ· (Γk f ) , ρ· (Γj Γk f )||W,α < ∞,
(3.9)
and the integral equation
Z
(3.10)
ρt (f ) = ν(f ) +
t
Z
ρs (Lf )ds +
0
t
ρs (Γk f )dWsk ,
0
holds.
Definition 15 (Forward RPDE solution). Given an α-Hölder rough path W = (W, W), α ∈
(1/3, 1/2], and φ ∈ Cb2 (Rd ) we say that p ∈ Cb0,2 ([0, T ] × Rd ) is a regular solution to
(
dpt = L∗ pt dt + Γ∗k pt dWtk
(3.11)
p0 = φ,
if Γ∗k p is controlled by W with Gubinelli derivative Γ∗j Γ∗k p and the integral equation
Z t
Z t
pt = φ +
L∗ ps ds +
Γ∗k ps dWtk ,
0
0
holds.
Theorem 16. Throughout, W is a geometric α-Hölder rough path, α ∈ (1/3, 1/2]. Assume σi , βj ∈
Cb3 (Rd ), b ∈ Cb1 (Rd ), c ∈ Cb1 (Rd ), γk ∈ Cb2 (Rd ). Let ν be a finite measure.
(i) Stability. Let ρ = ρW be the solution to (3.3) as given by the Feynman-Kac representation
(3.2), whenever W ∈ C 1 . Pick W ∈ C 1 convergent in rough path sense to W. Then
there exists a continuous finite-measure-valued function ρW , independent of the choice of the
approximating sequence, so that d(ρW , ρW ) → 0. The resulting map W 7→ ρW is continuous.
Moreover, the following Feynman-Kac representation holds for f ∈ Cb (Rd )
Z t
Z t
W
0,ρ0
ρt (f ) := E
f (Xt ) exp
c (Xs ) ds +
γ (Xs ) dWs ,
0
where X solves the same rough SDE as in Theorem 9.
0
SPDES AND ROUGH PATHS
15
(ii) Weak RPDE solution. The measure-valued path ρW constructed in part (i) is a solution to
(3.8) in the sense of Definition 14. Moreover, (3.9) is bounded, uniformly over bounded sets
of f in Cb3 (Rd ). If the coefficients satisfy the stronger conditions of Theorem 9 (iii) then ρW
is the only solution in the class of measure-valued functions ρ satisfying this uniform bound
on (3.9).
(iii) Regular RPDE solution. Assume σi ∈ Cb6 (Rd ), βj ∈ Cb7 (Rd ), b ∈ Cb5 (Rd ), γk ∈ Cb6 (Rd ), c ∈
4
Cb4 (Rd ). If ρ0 has a density p0 ∈ Cexp
(Rd ), then ρt has a density pt for all times, and
0,4
d
p ∈ Cexp ([0, T ] × R ) is a solution to (3.11) in the sense of Definition 15. It is the only
solution that in addition satisfies for some δ > 0
||Γ∗ u(·, x), Γ∗ Γ∗ u(·, x)||W,α . e−δ|x| .
Proof. (i): First of all we note that for fixed f ∈ Cb (Rd ) and fixed t we have that
W 7→ ρW
t (f )
is continuous in rough path topology. Indeed, this follows from Lemma 34 and is also seen to
hold uniformly in t and in bounded sets of f in Cb (Rd ). This also immediately gives the stated
Feynman-Kac representation.
(ii): Fix f ∈ Cb3 (Rd ) and for simplicity take b = γ = c = 0. Then note that
Z t
Z t
Z t
f (Xt ) = f (Xs ) +
Lf (Xr )dr +
hσi (Xr ), Df (Xr )idBri +
Γi f (Xr )dWri .
s
s
s
Taking expectation and applying Lemma 17 we get
Z t
Z t
ρt (f ) = ρs (f ) +
ρr (Lf )dr +
ρr (Γi f )dWri ,
s
s
as well as the desired uniform bound on (3.9).
To show uniqueness in part (ii), let φ ∈ Cb0,3 ([0, t], Rd ) be such that
Z t
Z t
φ(s, x) = ϕ(x) +
α(r, x)dr +
ηi (r, x)dWri ,
s
for some
0
(ηi=1,...,e , ηi,j=1,...,e
)
s
controlled by W , uniformly over x, i.e.
sup [||η(x), η 0 (x)||W,α ] < ∞,
x
sup [||Γi φ(x), Γi η(x)||W,α ] < ∞,
i = 1, . . . , e.
x
Moreover assume that η ∈ Cb0,3 ([0, T ] × Rd ). Then by Lemma 30
Z t
Z t
Z t
Z t
k
ρt (φt ) = ρ0 (φ0 ) +
ρr (Lφr )dr +
ρr (Γk φr )dWr −
ρr (α(r))dr −
ρr (ηk (r))dWrk .
0
0
0
0
0
So it remains to find, for given ϕ, such a φ with α(r) = Lφ(r), ηi (r) = Γi φ(r) and ηi,j
(r) =
4
d
Γj Γi φ(r). But this is exactly what Theorem 9 (iii) gives us for ϕ ∈ Cb (R ). Then
ρt (ϕ) = ρt (φt ) = ρ0 (φ0 ),
which gives uniqueness of ρ.
16
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
(iii): The coefficients of the adjoint equation are given in (1.4). In particular σ̃i ∈ Cb6 (Rd ), β̃j ∈
∈ Cb4 (Rd ), c̃ ∈ Cb4 (Rd ), γ̃k ∈ Cb6 (Rd ). Hence the adjoint equation fits into the setting of
Theorem 9 (iii). In particular there exists a Cb0,4 solution to (3.11) and we can represent it as
"
!#
Z
Z
Cb6 (Rd ), b̃
T
pt (x) = E p0 (X̃TT −t,x,W ) exp
T
c̃(XrT −t,x,W )dr +
γ̃(XrT −t,x,W )dWr
T −t
,
T −t
for a rough SDE X̃. The exponential estimates on the control then follow by a similar argument as
in the proof of Theorem 9 (iii), using Lemma 36 on the integrands Dg, D2 g. Finally, pt is the density
of ρt of part (ii) due to the following reason: pt is integrable because of the exponential decay, the
corresponding measure satisfies (3.11), which by uniqueness for that equation then coincides with
ρ.
The following lemma was needed in the previous proof.
Lemma 17. Assume σi , βj ∈ Cb3 (Rd ), b ∈ Cb1 (Rd ). Let Z be the joint lift of a Brownian motion B
with a (deterministic) geometric α-Hölder rough path W, α ∈ (1/3, 1/2] (see Lemma 33). Let X
be the random RDE solution to
Z t
Z t
Xt = X0 +
b(Xr )dr +
(σ, β) dZ
0
0
Z t
Z t
Z t
i
” = ”X0 +
b(Xr )dr +
σi (Xr )dBr +
βi (Xr )dWri .
0
Let f ∈
Cb3 (Rd )
0
0
and define
(Yt )i := E[βiz (Xt )∂k f (Xt )]
(Yt0 )i,j := E[∂k [∂z f βiz ](Xr )βjk (Xr )].
2α
and
Then (Y, Y 0 ) ∈ DW
Z
(3.12)
E[f (Xt )] = E[f (X0 )] +
t
Z
E[bf (Xr )]dr +
0
t
(Y, Y 0 )dWri .
0
Moreover for all R > 0,
sup
||(Y, Y 0 )||W,α < ∞.
||f ||C 3 ≤R
b
Proof. For simplicity take b = 0. First
i
i
Xs,t = σi (Xs )Bs,t
+ βi (Xs )Ws,t
+ Rs,t ,
3
where ||R||2α ≤ C (1 + ||Z||α ) (see Lemma 22).
Then, with gi := βiz ∂z f
j
j
gi (X)s,t = σjk (Xs )∂k gi (Xs )Bs,t
+ βjk (Xs )∂k gi (Xs )Ws,t
+ Rs,t ,
3
with ||βjk (Xs )∂k g(Xs )||α + ||R||2α ≤ C (1 + ||Z||α ) (see Lemma 28).
Taking expectation and using integrability of Z (Lemma 33), we get
Ys,t = Ys0 Ws,t + R̄s,t ,
with ||Y 0 ||α + ||R̄||2α ≤ C < ∞, as desired.
SPDES AND ROUGH PATHS
17
Now for W smooth, equation (3.12) is satisfied by Fubini’s theorem. Showing it for W a geometric
rough path then follows via smooth approximations. This has for example already been done - in
a similar setting - in the proof of Theorem 9, so we omit the details here.
The following result in the proof of the previous theorem is worth to be formulated separately.
Corollary 18 (Duality). Assume the conditions of Theorem 9 (iii). Let u be the unique solution
to the backward equation (2.7) and ρ be the unique solution to the forward equation (3.10). Then
ρt (ut ) = ρ0 (u0 )
∀t ∈ [0, T ].
4. Appendix
4.1. Rough differential equations. We recall the space of controlled paths.
Definition 19. Let X be an α-Hölder continuous path, α ∈ (1/3, 1/2]. A path Y ∈ C α is controlled
2α
by X with derivative Y 0 ∈ C α (in short (Y, Y 0 ) ∈ DW
), if
Y
Ys,t = Ys0 Xs,t + Rs,t
,
2α
,
with RY = O(|t − s|2α ). We use the following semi-norm on DW
||Y, Y 0 ||X,α := ||Y 0 ||α + ||RY ||2α .
Definition 20. Let ω be a control function (see Definition 1.6 in [14]). For a > 0 and [s, t] ⊂ [0, T ]
we set
τ0 (a)
τi+1 (a)
= s
=
inf {u : ω (τi , u) ≥ a, τi (a) < u ≤ t} ∧ t
and define
Na,[s,t] (ω) = sup {n ∈ N∪ {0} : τn (a) < t} .
p
When ω arises from a (homogeneous) p-variation norm of a (p-rough) path, such as ωX = kXkp-var;[·,·] ,
we shall also write
Na,[s,t] (X) := Na,[s,t] (ωX ) .
Remark 21. The importance of Na;[0,T ] (X) stems from the fact that it has - contrary to the pvariation norm ||X||p−var - Gaussian integrability if X = B, the lift of Brownian motion (see
[6, 13]), or if X = Z, the joint lift of Brownian motion and a deterministic rough path used in the
proof of Theorem 9 (see Lemma 33 (iii)).
Lemma 22 (Bounded vector fields). Let X be a geometric α-Hölder rough path, α ∈ (1/3, 1/2].
Let
dY = V (Y )dX,
where V = (Vi )i=1,...,e is a collection of Lip3 (Rd ) vector fields.
Then with (Y, Y 0 ) := (Y, V (Y )) we have
(4.1)
3
||Y, Y 0 ||X,α ≤ C (1 + ||X||α ) .
Also
(4.2)
||Y ||1/α−var ≤ C 1 + N1;[0,T ] (X) .
Proof. (4.1) follows from [19, Proposition 8.3] and (4.2) follows from [13, Lemma 4, Corollary 3]. 18
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
Lemma 23 (Linear vector fields). Let X be a geometric α-Hölder rough path, α ∈ (1/3, 1/2]. Let
dY = V (Y )dX,
where V = (Vi )i=1,...,e is a collection of linear vector fields of the form Vi (z) = Ai z + bi , where Ai
are d × d matrices and bi ∈ Rd . Then for 0 ≤ s ≤ t ≤ T :
(1)
||Y ||p−var;[s,t] ≤ C (1 + |y0 |) ||X||p−var;[s,t] exp(CN1;[0,T ] (X)).
with p := 1/α, which implies
|Y |α;[0,T ] ≤ C (1 + |y0 |) ||X||α;[0,T ] exp(CN1;[0,T ] (X)).
(2)
|Ys,t − V (Ys )Xs,t − DV (Ys )V (Ys )Xs, | ≤ C exp(CN1 )||X||3p−var;[s,t] ,
which means that with (Y, Y 0 ) := (Y, V (Y )) we have
||Y, Y 0 ||X,α ≤ C exp(CN1 ) ||X||2α;[0,T ] ∨ ||X||3α;[0,T ] .
Proof. 1. In what follows C is a constant that can change from line to line.
From [14, Theorem 10.53] we have for any s ≤ u ≤ v ≤ t:
||Yu,v || ≤ C (1 + |Yu |) ||X||p−var;[u,v] exp(c||X||pp−var;[u,v] ).
Then, using ||Yu,v || = d(Yu , Yv ) ≥ d(Ys , Yv ) − d(Ys , Yu ) = ||Ys,v || − ||Ys,u ||, we have
||Ys,v || ≤ C (1 + |Yu |) ||X||p−var;[u,v] exp(C||X||pp−var;[u,v] ) + ||Ys,u ||
≤ C (1 + |Ys | + ||Ys,u ||) ||X||p−var;[u,v] exp(C||X||pp−var;[u,v] ) + ||Ys,u ||.
(4.3)
On the one hand, this gives
||Ys,v || ≤ C (1 + |Ys | + ||Ys,u ||) exp(C||X||pp−var;[u,v] ).
Now letting s = τ0 < · · · < τM < τM +1 = v ≤ t, by induction,
||Ys,v || ≤ C M +1 ((M + 1)(1 + |Ys |)) exp(C
M
X
||X||pp−var;[τi ,τi+1 ] )
i=0
≤ C M +1 (1 + |Ys |) exp(C
M
X
||X||pp−var;[τi ,τi+1 ] ).
i=0
Hence
sup ||Ys,u || ≤ C (1 + |Ys |) exp(CN1;[s,t] ).
u∈[s,t]
Then, using again (4.3),
||Ys,v ||
≤ C (1 + |Ys | + ||Ys,u ||) ||X||p−var;[u,v] exp(C||X||pp−var;[u,v] ) + ||Ys,u ||
≤ C 1 + |Ys | + C (1 + |Ys |) exp(CN1;[s,t] ) ||X||p−var;[u,v] exp(C||X||pp−var;[u,v] ) + ||Ys,u ||
≤ C 2 2 (1 + |Ys |) ||X||p−var;[u,v] exp(C||X||pp−var;[u,v] ) exp(CN1;[s,t] ) + ||Ys,u ||.
SPDES AND ROUGH PATHS
19
Then letting s = τ0 < · · · < τM < τM +1 = v ≤ t, by induction,
||Ys,v ||
≤
M
X
C 2 2 (1 + |Ys |) ||X||p−var;[τi ,τi+1 ] exp(C||X||pp−var;[τi ,τi+1 ] ) exp(CN1;[s,t] )
i=0
≤
M
X
C 2 2 (1 + |Ys |) ||X||p−var;[s,v] exp(C||X||pp−var;[τi ,τi+1 ] ) exp(CN1;[s,t] )
i=0
≤ (M + 1)C (1 + |Ys |) ||X||p−var;[s,v] exp(C max ||X||pp−var;[τi ,τi+1 ] ) exp(CN1;[s,t] ).
i
Then
||Ys,t || ≤ (N1;[s,t] + 1)C (1 + |Ys |) exp(CN1;[s,t] )||X||p−var;[s,t]
≤ C (1 + |Y0 |) exp(CN1;[0,T ] )||X||p−var;[s,t] ,
as desired.
2. It is straightforward to construct Lip3 vector fields Ṽi , that coincide with Vi on an open
neighborhood of Y and they can be chosen in such a way that
||Ṽ ||Lip3 ≤ max (|Ai | + |bi |) (|Y |∞ + 2) ≤ C exp(CN1 ).
i
The first statement then follows from [14, Corollary 10.15], which also yields the desired bound on
||RY ||2α . The bound on ||Y 0 ||α follows from Step 1.
Lemma 24. Let ϕ ∈ Cb2 (Rd , R), X a geometric β-Hölder rough path in Re , β ∈ (1/3, 1/2],
V1 , . . . , Ve ∈ Cb3 (Rd ), and Ψ the flow to the RDE
dY = V (Y )dX.
Then
−1
−1
−1 )
det(DΨ
)
−
ϕ(Ψ
)
det(DΨ
)
ϕ(Ψ−1
t,T
t,T
s,T
s,T 17+3d
(4.4)
≤ C||ϕ||Cb2 (M (y)) exp(CN1;[0,T ] (X)) (||X||β + 1)
|t − s|β ,
−1
−1
−1
−1
−1
)
det(DΨ
)
−
ϕ(Ψ
)
det(DΨ
)
−
div(ϕV
)(Ψ
)
det(DΨ
)X
ϕ(Ψ−1
s,t
t,T
t,T
s,T
s,T
t,T
t,T
≤ C||ϕ||Cb2 (M (y)) exp(CN1;[0,T ] (X)) (||X||β + 1)
17+3d
|t − s|2β ,
with C = C(β, V, ϕ). Here the inverse flow and its Jacobian are evaluated at y ∈ Rd . Moreover we
used
−1
M (y) := {x : inf |Ψ−1
T −r,T (y)| − 1 ≤ |x| ≤ sup |ΨT −r,T (y)| + 1}.
r∈[0,T ]
r∈[0,T ]
Proof. We shall need the fact that the inverse flow and its Jacobian satisfy the following RDEs (see
for example [14, Section 11]),
(4.5)
−1
dΨ−1
T −·,T (y) = V (ΨT −·,T (y))dXT −· ,
Ψ−1
T,T (y) = y,
(4.6)
−1
−1
i
dDΨ−1
T −·,T (y) = DVi (ΨT −·,T (y))DΨT −·,T (y)dXT −· ,
DΨ−1
T,T (y) = I,
We proceed to show the second inequality of the statement, as the first one follows analogously.
In what follows C will denote a constant changing from line to line, only depending on β, V , ϕ (but
not on X or y).
20
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
2β
−1
0
Let Ar := ϕ(Ψ−1
− , with
T −r,T (y)), Br := det(DΨT −r,T (y)). Using (4.5) we have that (A, A ) ∈ D←
X
A0r
hDϕ(Ψ−1
T −r,T (y)), V
(Ψ−1
T −r,T (y))i
=
←
−
and X r := XT −r . More specific, by Lemma 22 together with Lemma 28
8
−
≤ C||ϕ||Cb2 (M (y)) (||X||α + 1) .
||A||←
X ,β
−1
where M (y) := {x : inf r∈[0,T ] |Ψ−1
T −r,T (y)| − 1 ≤ |x| ≤ supr∈[0,T ] |ΨT −r,T (y)| + 1}.
Moreover using (4.6) and the derivative of the determinant,
D det |A · M = det(A) Tr[A−1 M ],
(4.7)
2β
we get that (B, B 0 ) ∈ D←
− , with
X
−1
Br0 = (div V )(Ψ−1
T −r,T (y)) det(DΨT −r,T (y)).
More specific, by Lemma 23.2 together with Lemma 28
−
||B||←
≤ C|| det ||Cb2 (N (y)) exp(CN1;[0,T ] ) (||X||α + 1)
X ,β
!d
≤C
1 + sup |DΨ−1
T −r,T (y)|
exp(CN1;[0,T ] ) (||X||α + 1)
8
8
r∈[0,T ]
where N (y) := {A : |A| ≤ supr∈[0,T ] |DΨ−1
T −r,T (y)| + 1}.
Applying Lemma 26, we get for 0 ≤ u < v ≤ T
←
− Av Bv − Au Bu − (A0u Bu − Au Bu0 ) X u,v 8
≤ C (1 + ||X||α ) |ϕ(y)| + ||ϕ||Cb2 (M (y)) (||X||α + 1)

!d
× 1 +
1 + sup |DΨ−1
T −r,T (y)|

8
exp(CN1;[0,T ] ) (||X||α + 1)  ,
r∈[0,T ]
Finally noting
−1
A0r Br + Ar Br0 = − div(ϕV )(Ψ−1
T −r,T (y)) det(Dϕ(ΨT −r,T (y)),
and using t = T − u, s = T − t, the desired result follows from Lemma 27.
The following result from the previous proof is worth noting separately.
Lemma 25 (Liouville’s formula for RDEs). Let X be a matrix-valued, geometric α-Hölder rough
path, α ∈ (0, 1] and consider the matrix-valued linear equation
dMt = dXit · Mt
M0 = I ∈ Rd×d .
Denote Dt := det(Mt ), then
dDt = Dt tr[dZti ],
D0 = 1,
SPDES AND ROUGH PATHS
21
which is explicitly solved as
Dt = exp(tr[Xti ] − tr[X0i ]).
2α
2α
Lemma 26. Let X ∈ C α , and (A, A0 ), (B, B 0 ) ∈ DX
. Then (Y, Y 0 ) := (AB, A0 B + AB 0 ) ∈ DX
and
||Y 0 ||α + ||RY ||2α ≤ C (1 + ||X||α ) (|A0 | + ||A, A0 ||X,α ) (|B0 | + ||B, B 0 ||X,α ) .
Proof. Straightforward calculation.
2α
2α
Lemma 27. Let Ỹt := YT −t , Ỹt0 := YT0 −t , X̃t := XT −t . If (Ỹ , Ỹ 0 ) ∈ DX̃
, then (Y, Y 0 ) ∈ DX
and
||Y 0 ||α + ||RY ||2α ≤ ||Ỹ 0 ||α + ||RỸ ||2α + ||Ỹ 0 ||α ||X||α .
Proof. This follows from
0
Yt − Ys − Ys0 Xs,t = Yt − Ys − Yt0 Xs,t + Ys,t
Xs,t
0
= Ỹu − Ỹv − Ỹu0 Xv,u + Ys,t
Xs,t
0
= − Ỹv − Ỹu − Ỹu0 Xu,v + Ys,t
Xs,t ,
where v := T − s, u := T − t.
2α
Lemma 28. Let X be an α-Hölder path, α ∈ (1/3, 1/2]. Let (Y, Y 0 ) ∈ DX
, φ ∈ Cb2 . Then
0
2α
(φ(Y ), Dφ(Y )Y ) ∈ DX with
2
2
||φ(Y ), Dφ(Y )Y 0 ||X,α ≤ C(α, T )||φ||Cb2 (1 + ||X||α ) (1 + |Y00 | + ||Y, Y 0 ||X,α ) .
Proof. See [19, Lemma 7.3].
Lemma 29 (Adjoint equation). Let X be a geometric α-Hölder rough path, α ∈ (1/3, 1/2]. Let
V = (Vi )i=1,...,e be a collection of Lip3 (Rd ) vector fields. Let
dYst,x = V (Yst,x )dXs
Ytt,x = x.
Then
dYTt,x = −DYTt,x V (x)dXt
dDYTt,x = −D2 YTt,x hV (x)dXt , ·i − DYTt,x DV (x)dXt .
Proof. Take the time derivative of
t,YT−1,t,x
YT
= x,
for the first identity and consider the enlarged equation
dZ = G(Z)dX,
with G(x1 , x2 ) = (V (x1 ), DV (x1 )x2 ) for the second identity.
22
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
Lemma 30. Let α ∈ (1/3, 1/2] and W a geometric α-Hölder rough path. Let ρ be a solution to
the forward equation (3.8) in the sense of Definition 14; in particular (ρt (f ), ρt (Γf )) is controlled
for every f ∈ Cb3 . Assume moreover that for every R > 0
sup
{||ρ· (f ), ρ· (Γf )||W,α } < ∞.
||f ||C 3 (Rd ) <R
b
Let φ ∈ Cb0,3 ([0, T ] × Rd ) be given, satisfiying for s ≤ t
Z t
Z t
φ(t, x) = φ(s, x) +
αr (x)dr +
(ηr (x), ηr0 (x))dWr ,
s
where
(ηt (x), ηt0 (x))
∈
2α
DW
.
s
Assume moreover
sup [||η(x), η 0 (x)||W,α ] < ∞,
x
sup [||Γi φ(x), Γi η(x)||W,α ] < ∞, i = 1, . . . , e.
x
and η ∈ Cb0,3 ([0, T ] × Rd ).
2α
where
Then (M, M 0 ), (N, N 0 ) ∈ DW
(Mt )i=1,...,e := ρt (Γi φt )
(Mt0 )i,j=1,...,e := ρt (Γj Γi φt ) + ρt (Γi (ηt )j )
(Nt )i=1,...,e := ρt ((ηt )i )
(Nt0 )i,j=1,...,e := ρt ((ηt0 )ij ) + ρt (Γj (ηt )i ),
and
Z
ρt (φt ) = ρs (φs ) +
t
(M, M 0 )r dWr +
s
Z
t
(N, N 0 )r dWr +
Z
s
t
ρr (αr + Lφr )dr.
s
Remark 31. Note that with (η, η 0 ) ≡ 0, α ≡ 0, φ(0, x) = f (x), this reduces to (3.10).
Proof. First
(Mt )i − (Ms )i = ρt (Γi φt ) − ρs (Γi φs )
= ρs (Γi φs,t ) − ρs,t (Γi φs ) + ρs,t (Γi φs,t )
j
j
= ρs (Γi (ηs )j )Ws,t
− ρs (Γj Γi φs )Ws,t
+ O(|t − s|2α ).
j
Here we used that by assumption Γi φs,t (x) = Γi ηj (x)Ws,t
+ O(|t − s|2α ), uniformly in x and that
j
2α
ρs,t (Γi f ) = ρs (Γj Γi f )Ws,t + O(|t − s| ) uniformly over bounded sets of f in Cb3 . It follows that
2α
(M, M 0 ) ∈ DW
. And analogously for (N, N 0 ):
(Nt )i − (Ns )i = ρt ((ηt )i ) − ρs ((ηs )i )
= ρs ((ηs,t )i ) − ρs,t ((ηs )i ) + ρs,t ((ηs,t )i )
j
j
= ρs ((ηs0 )i,j Ws,t
+ ρs (Γj (ηs )i )Ws,t
+ O(|t − s|2α ),
since ηt ∈ Cb3 (Rd ) uniformly in t.
SPDES AND ROUGH PATHS
23
Now for the integral equality, for simplicity take L = 0, α = 0. Then
ρt (φt ) − ρs (φs )
= ρs (φs,t ) + ρs,t (φs ) + ρs,t (φs,t )
ij
j
i
i
i
= ρs ((ηs )i )Ws,t
+ ρt ((ηs0 )ij )Wij
s,t + ρs (Γi φs )Ws,t + ρs (Γj Γi φs )Ws,t + ρs (Γi (ηs )j )Ws,t Ws,t
+ O(|t − s|3α )
ij
i
i
= ρs ((ηs )i )Ws,t
+ [ρt ((ηs0 )ij ) + ρs (Γj (ηs )i )] Wij
s,t + ρs (Γi φs )Ws,t + [ρs (Γj Γi φs ) + ρs (Γi (ηs )j )] Ws,t
+ O(|t − s|3α ).
Now for every partition P of [s, t] we have the trivial identity
X
ρt (φt ) − ρs (φs ) =
[ρv (φv ) − ρu (φu )] .
[u,v]∈P
The claimed equality then follows from taking the limit along partitions with mesh-size going to
zero.
Lemma 32. Let W be a geometric α-Hölder rough path, α ∈ (1/3, 1/2]. Let u be a weak solution
to the backward RPDE (2.5) in the sense of Definition 4; in particular (hu· , Γ∗ φi, hu· , Γ∗ Γ∗ φi)) is
3
(Rd ). Assume moreover that for every R > 0
controlled for every φ ∈ Cexp
sup
(4.8)
{||(hu· , f i, hu· , Γf i)||W,α } < ∞.
||f ||C 3
d <R
exp (R )
0,4
Let φ ∈ Cexp
([0, T ] × Rd ) be given that satisfies for s ≤ t
Z t
Z t
φ(t, x) = φ(s, x) +
αr (x)dr +
(ηr (x), ηr0 (x))dWr ,
s
where
(ηt (x), ηt0 (x))
∈
2α
DW
.
s
Assume moreover for some δ > 0
||η(x), η 0 (x)||W,α . e−δ|x| ,
||Γ∗i φ(x), Γ∗i η(x)||W,α . e−δ|x| , i = 1, . . . , e.
0,3
In addition assume that η ∈ Cexp
([0, T ] × Rd ).
0
0
2α
Then (M, M ), (N, N ) ∈ DW where
(Mt )i=1,...,e := hut , Γ∗i φt i
(Mt0 )i,j=1,...,e := hut , Γ∗j Γ∗i φt i + hut , Γ∗i (ηt )j i
(Nt )i=1,...,e := hut , (ηt )i i
(Nt0 )i,j=1,...,e := hut , (ηt0 )ij i + hut , Γ∗j (ηt )i i,
and
Z
hut , φt i = hus , φs i −
s
t
(M, M 0 )r dWr +
Z
s
t
(N, N 0 )r dWr +
Z
t
hur , αr − L∗ φr idr.
s
Proof. By assumption
|Γ∗i φt (x) − Γ∗i φs (x) − Γ∗i ηs (x)Ws,t | . e−δ|x| |t − s|2α
24
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
Hence
j
|hus , Γ∗i φs,t i − hus , Γ∗i ηj iWs,t
| ≤ ||u||∞ ||β||Cb2 (Rd ) ||Dφt − Dφs − Dηs Ws,t ||L1 (Rd ) . |t − s|2α .
Moreover
j
|hus,t , Γ∗i φs i − hus , Γ∗j Γ∗i φs iWs,t
| . |t − s|2α ,
0,4
3
since φ ∈ Cexp
([0, T ] × Rd ) (and hence Γ∗j ut ∈ Cexp
(Rd ) uniformly in t) and since u satisfies the
∗
uniform bound (4.8). Then also hus,t , Γi φs,t i . |t − s|2α and hence
j
j
(Mt )i − (Ms )i = hut , Γ∗j Γ∗i φt iWs,t
+ hut , Γ∗i (ηt )j iWs,t
+ O(|t − s|2α ),
hence (M, M 0 ) is controlled. The argument for (N, N 0 ) is similar and the proof now finishes as the
preceding one.
4.2. Rough SDEs.
Lemma 33. For W a geometric α-Hölder rough path, α ∈ (1/3, 1/2] and a Brownian motion B,
define Z = (Z, Z) as
R
Ito
Bs,t
W ⊗ dB
Bt
Rt
Zt =
, Zs,t =
Wt
Bs,t ⊗ dW W
s
Then
(i) Z is well-defined and, almost surely, an α-Hölder rough path
(ii)
ρα Z(W), Z(W̃) . ρα W, W̃ .
Lq
(iii)
N1;[0,T ] (||Z||pp )
has Gaussian tails, uniformly over W bounded, for all p =
1
α.
Proof. This is proven in [11], the only difference being that there, Z is only shown to be an α0 Hölder rough path, for α0 < α. This stems from the fact, that there, a Kolmogorov-type argument
is applied to the whole rough path Z, which in particular contains the deterministic path W , which
explains the decay in perceived regularity.
Being more careful, and applying a Kolmogorov-type argument (e.g. Theorem 3.1 in [19]) only
to the second level, one sees that it is actually β-Hölder continuous, for β < α + 1/2. The first level
is trivially α-Hölder continuous. The claimed continuity in W is then improved similarly.
Lemma 34 (Rough SDE). Le W be a geometric α-Hölder rough path, α ∈ (1/3, 1/2] and let
Z = (Z, Z) be the joint lift of W and a Brownian motion B, given in the previous Lemma 33.
Assume σi , βj ∈ Cb3 (Rd ), i = 1, . . . , dB , j = 1, . . . , e, b ∈ Cb1 (Rd ). Let X = X(ω) be the solution to
the rough differential equation
dX = b(X)dt + V (X)dZ,
where V = (σ, β). Then X formally solves the rough SDE
dX = b(X)dt + σ(X)dB + β(X)dW.
We have the following properties:
SPDES AND ROUGH PATHS
25
• For all p ≥ 1, the mapping
C α → Sp
W 7→ X,
is locally uniformly continuous. Here ||X||pS p := E[supt≤T |Xt |p ]. Moreover for every R > 0
there is δ > 0 such that
sup
||W||α <R
E[exp(δ|X W |2∞ )] < ∞.
If in addition, for n ≥ 0, σi , βj ∈ Cbη+n , b ∈ Cb1+ε+n , then the same holds true for Dn X.
• For W the canonical lift of a smooth path W , X coincides with the classical SDE solution
to
dXt = b(Xt )dt + σ(Xt )dB + β(Xt )Ẇt dt.
(4.9)
• Let c ∈ Cb1 (Rd ), γ ∈ Cbη (Rd ) and g ∈ Cb1 (Rd ), then
integral, and moreover, for all p ≥ 1
R
γ(Xs )dWs is a well-defined rough
[0, T ] × Rd × C α → Lp
(t, x, W) 7→
Z
g(XTt,x ) exp
T
c(Xrt,x )dr
Z
γ(Xrt,x )dWr
+
t
!
T
,
t
is continuous, uniformly in t, x and locally uniformly in W.
Proof. If b ∈ Cb1+ε (Rd ), some ε > 0 this is shown in [11, Theorem 10]. Now, for b ∈ Cb1 (Rd ) the
same proof works, one just needs to use the improved result on RDEs with drift in [15, Proposition
3].
Lemma 35. Let W be a geometric α-Hölder rough path, α ∈ (1/3, 1/2]. Let X t,x be the solution
to the rough SDE (Lemma 34)
Xtt,x = x.
dXt = b(Xt )dt + σ(Xt )dB + β(Xt )dW,
3+[(n−1)∨0]
Let n ≥ 0 and assume c ∈ Cbn (Rd ), γk ∈ Cb2+n (Rd ), σi , βj ∈ Cb
n
Then for every φ ∈ Cexp
(Rd ) the function
ψ(x) :=
E[φ(XTt,x ) exp
Z
T
c
t
Xrt,x
Z
dr +
1+[(n−1)∨0]
(Rd ), b ∈ Cb
!
T
γ
Xrt,x
dWr ],
t
n
is again in Cexp
(Rd ), with ||ψ||Cexp
n (Rd ) bounded uniformly for t ≤ T and ||W||α bounded.
.
26
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
Proof. For n = 0, let C1 > 0 such that |ψ(x)| ≤ C1 exp(− C11 |x|). Then
!
Z T
Z T
t,x
t,x
t,x
γ Xr dWr ]
c Xr dr +
|ψ(x)| = E[φ(XT ) exp
t
t
1
≤ C1 E[exp(− |XTt,x |) exp(. . . )]
C1
1
1
≤ C1 exp(− x)E[exp( |XTt,x − x|) exp(. . . )]
C1
C1
1
≤ C1 exp(− x)E[exp(C2 1 + N1,[t,T ] (Z) + T ||c||∞ )]
C1
1
≤ C1 exp(− x)E[exp(C3 N1,[0,T ] (Z))],
C1
where we used (4.2) for the 4th line This concludes the argument, since the expectation is finite
by Lemma 33, uniformly for ||W||α bounded. The case n ≥ 1 follows similarly, by differentiating
under the expectation.
Lemma 36. Let W be a geometric α-Hölder rough path, α ∈ (1/3, 1/2]. Assume σi , βj ∈ Cb3 (Rd ), i =
1, . . . , dB , j = 1, . . . , e, b ∈ Cb1 . Let X t,x be the solution to the rough SDE (Lemma 34)
dX = σ(X)dB + β(X)W.
n
(Rd ), any q ≥ 1 the function
Let n ≥ 0. For every ϕ ∈ Cexp
ψ(x) := E[||ϕ||qC n (M (x)) ],
b
is in
0
Cexp
.
Here M (y) := {x :
inf r∈[t,T ] |Xrt,x |
− 1 ≤ |x| ≤ supr∈[t,T ] |Xrt,x | + 1}.
Proof. This follows from
||ϕ||L∞ (M (x)) ≤ C exp(−δ inf |Xrt,x |)
r∈[t,T ]
≤ C exp(−δ inf |Xrt,x |)
r∈[t,T ]
≤ C exp(−δ|x|) exp(δ sup |Xrt,x − x|).
r∈[t,T ]
Acknowledgement:
All authors acknowledge support from DFG Priority Program 1324. P. Friz received funding
from the European Research Council under the European Union’s Seventh Framework Programme
(FP7/2007-2013) / ERC grant agreement nr. 258237.
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28
JOSCHA DIEHL, PETER K. FRIZ AND WILHELM STANNAT
JD and WS are affiliated to TU Berlin. PKF is corresponding author ([email protected])
and affiliated to TU and WIAS Berlin.